[Paper Review] Magnon frequency renormalization by the electronic geometrical spin torque in itinerant magnets

Emil Viñas Boström, F. G. Eich+1 more|arXiv (Cornell University)|Dec 13, 2021

Magnetic properties of thin films4 citations

TL;DR

This paper proposes a semi-classical model for magnon dynamics in itinerant magnets like Fe, Co, and Ni, showing that the electronic spin Berry curvature induces a geometrical torque that renormalizes magnon frequencies. The key result is that this non-adiabatic effect explains the anomalous magnon frequency hardening and spectral narrowing observed in first-principles calculations, particularly near a critical spin-electron coupling strength.

ABSTRACT

We investigate non-adiabatic effects on the magnon frequency in an interacting system of localized spins and itinerant electrons. Including the lowest order corrections to the adiabatic dynamics in an analytically solvable model, applicable to simple ferromagnets like Fe, Co and Ni, we find that the magnon frequency is renormalized by a geometrical torque arising from the electronic spin Berry curvature. Comparison to exact numerical simulations reveals that our analytical solution captures essential low-energy features, and provides a mechanism for the magnon frequency hardening observed in recent first principles calculations for Fe, provided the geometrical torque is taken into account.

Motivation & Objective

• To understand non-adiabatic corrections to magnon dynamics in itinerant magnets such as Fe, Co, and Ni.
• To address the failure of adiabatic approximations in capturing anomalous magnon frequency shifts observed in first-principles calculations.
• To develop an analytically solvable model that includes next-to-leading-order corrections in the adiabatic parameter ω/ε.
• To link the observed magnon frequency hardening to the electronic spin Berry curvature and its dynamical torque on localized spins.
• To provide a mechanism for spectral peak narrowing and frequency hardening in non-linear response regimes, consistent with TD-DFT results.

Proposed method

• Formulates a coupled spin-electron Hamiltonian with itinerant electrons and localized spins, using a semi-classical approximation valid for large spin S.
• Applies the Lagrangian formalism to derive an effective equation of motion for localized spins, incorporating non-adiabatic corrections.
• Identifies a geometrical torque in the effective spin equation arising from the electronic spin Berry curvature, which emerges in the adiabatic limit.
• Solves the resulting spin dynamics analytically to next-to-leading order in ω/ε, where ω and ε are the characteristic energy scales of spin and electron systems.
• Compares analytical results with exact numerical time-evolutions of the model and with real-time TD-DFT simulations for BCC Fe.
• Uses Fourier transforms of spin correlation functions to extract magnon frequencies and spectral features, comparing them to first-principles data.

Experimental results

Research questions

• RQ1 How do non-adiabatic effects from electronic spin Berry curvature influence magnon frequency in itinerant magnets?
• RQ2 Can a semi-classical model with geometrical torque explain the anomalous magnon frequency hardening seen in recent first-principles calculations of Fe?
• RQ3 What is the role of spin-flip excitations in the electronic system in mediating the geometrical torque on localized spins?
• RQ4 How does the magnon frequency depend on the spin-electron coupling strength g, and what causes the transition at g ≈ g_c?
• RQ5 Why do spectral peaks narrow with increasing perturbation strength in non-linear response, and is this captured by the proposed model?

Key findings

• The magnon frequency is renormalized by a geometrical torque arising from the electronic spin Berry curvature, which enters the effective spin equation of motion at next-to-leading order in the adiabatic expansion.
• The model correctly captures the anomalous hardening of the magnon frequency when the spin-electron coupling exceeds a critical value g_c ≈ 2 sin(q/2), matching first-principles TD-DFT results for Fe.
• Spectral narrowing with increasing perturbation strength is explained by the reduced rate of spin-flip excitations at higher θ, leading to longer magnon lifetimes.
• For g < g_c, the two-mode structure in the spin dynamics leads to nutational motion and a beat frequency ω = ω₁ − ω₂, with ω₂ ≈ δ corresponding to electronic spin-flip excitation energy.
• The semi-adiabatic approach fails to capture the second mode near g ≈ g_c but accurately reproduces the fundamental magnon frequency and its hardening across all g.
• The effect is robust and expected to occur in a broad class of magnetic materials due to the generic presence of non-zero spin Berry curvature in itinerant systems.